Reaction-Diffusion Patterns in Plant Tip Morphogenesis: Bifurcations on Spherical Caps

Abstract

We study a chemical reaction-diffusion model (the Brusselator) for pattern formation on developing plant tips. A family of spherical cap domains is used to represent tip flattening during development. Applied to conifer embryos, we model the chemical prepatterning underlying cotyledon ("seed leaf") formation, and demonstrate the dependence of patterns on tip flatness, radius, and precursor concentrations. Parameters for the Brusselator in spherical cap domains can be chosen to give supercritical pitchfork bifurcations of patterned solutions of the nonlinear reaction-diffusion system that correspond to the cotyledon patterns that appear on the flattening tips of conifer embryos.

Two-stage patterning dynamics in conifer cotyledon whorl morphogenesis

“…This depletion-type kinetics tends to form regular patterns in X 1 and Y 1 very close to predictions from linear analysis (e.g. Lacalli, 1981;Harrison, 1993;Holloway and Harrison, 1995) and has been used extensively for regular branching processes in morphogenesis (Harrison and Kolář, 1988;Harrison, 1999, 2008;Nagata et al, 2003Nagata et al, , 2013Jönsson et al, 2005;Rozada et al, 2013). Due to this regularity, the Brusselator was used to model formation of the P1 annulus ( Fig.…”

“…P2 patterning is disrupted in cup-shaped embryos. γ denotes the flatness of the embryo (as defined in Nagata et al, 2013 mechanisms can have very complex dynamics, because even subsets of the terms -such as reaction and diffusion (Turing, 1952), or diffusion and up-the-gradient PAT (e.g. Jönsson, et al, 2006) -are capable of self-organization.…”

Mapping nano-scale mechanical heterogeneity of primary plant cell walls

“…The biological implications of nano-scale mechanical heterogeneities remain unknown; we hypothesize that they may serve as an important prerequisite for development of mechanical zones and patterns which have a strong association with plant cell expansion; so-called reaction–diffusion patterns ( Nagata et al , 2013 ). In addition, the domain structure of mechanical properties may also be related to the distribution and clustering of microtubules ( Xiao et al , 2016 ).…”

A quantitative study of cotyledon positioning in conifer development

“…The Nagata et al (2013) analysis for the flattening dome indicated that, to maintain a particular n c , embryos would tend to grow in diameter as they flattened from a hemisphere to a disk. (Note: this analysis was for patterning of a single mechanism on the whole embryo surface, i.e.…”

“…1d). To more precisely characterize the patterning modes available on the embryo geometry, Nagata et al (2013) analyzed the harmonics for a reaction-diffusion mechanism on a flattening dome, specified by a parameter γ (Fig. 1e) varying between 1 (hemispherical dome) and 0 (flat disk).…”